Problem 49 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$3 x+2 y=6$$

Step-by-Step Solution

Answer

The points are (0, 3) and (2, 0); the slope is \(-\frac{3}{2}\).

Step 1: Rearrange the Equation into Slope-Intercept Form

The slope-intercept form of a line equation is given by \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Start by rearranging the line equation \(3x + 2y = 6\) to solve for \(y\):\(2y = -3x + 6\)Divide every term by 2 to isolate \(y\):\[y = -\frac{3}{2}x + 3\]

Step 2: Find Two Points on the Line

To find two points, choose values for \(x\) and solve for \(y\). Let's choose \(x = 0\) and \(x = 2\):1. When \(x = 0\): \(y = -\frac{3}{2}(0) + 3 = 3\) So one point is \((0, 3)\).2. When \(x = 2\): \(y = -\frac{3}{2}(2) + 3 = -3 + 3 = 0\) So another point is \((2, 0)\).

Step 3: Calculate the Slope Using Two Points

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Using the points \((0, 3)\) and \((2, 0)\):\[m = \frac{0 - 3}{2 - 0} = \frac{-3}{2}\]Thus, the slope of the line is \(-\frac{3}{2}\).

Key Concepts

Understanding Coordinates

Coordinates are used to specify the location of points on a plane. In algebra, when we talk about coordinates, we typically refer to a pair of numbers:

• The first number is the x-coordinate, which represents a point's horizontal position from the origin (0,0) on the x-axis.
• The second number is the y-coordinate, indicating the point's vertical position on the y-axis.

For example, in the ordered pair (0, 3), '0' is the x-coordinate, and '3' is the y-coordinate. This tells us the position of a point relative to the origin. By selecting pairs of coordinates, such as (0, 3) and (2, 0), we can identify and plot precise locations of points. These points can then be used in further calculations, such as determining the slope of the line that connects them. Understanding how to work with coordinates is foundational for graphing equations and finding relationships between points.

Exploring the Slope-Intercept Form

In algebra, one way to express the equation of a line is through the slope-intercept form, given as:\[y = mx + b\]Here, 'm' represents the slope of the line, and 'b' is the y-intercept, which is where the line crosses the y-axis. To convert a standard line equation like \(3x + 2y = 6\) into this form, we solve for 'y':

• First, isolate terms involving 'y': \(2y = -3x + 6\).
• Then, divide every term by 2: \(y = -\frac{3}{2}x + 3\).

This is now in slope-intercept form, where the slope \(m\) is \(-\frac{3}{2}\) and the y-intercept \(b\) is 3. This form is beneficial because it immediately shows us the slope and where the line intercepts the y-axis, providing a clear method to visualize and draw the line on a graph.

Calculating the Slope Between Two Points

The slope of a line represents how steep the line is, essentially measuring the vertical change versus the horizontal change between two points. The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Using the points from our exercise,

• \((0, 3)\)
• \((2, 0)\)

we can calculate the slope: \[m = \frac{0 - 3}{2 - 0} = \frac{-3}{2} \]This means that for each unit increase in the x-direction, the line decreases by 1.5 units in the y-direction. Calculating the slope is crucial because it helps understand the line's direction and steepness. In every linear function, calculating slope accurately ensures correct plotting and analysis of relations between variables involved.